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Fighter Aircraft AvionicsPart IV
SOLO HERMELIN
Updated: 04.04.13
1
Table of Content
SOLO Fighter Aircraft Avionics
2
Introduction
Jet Fighter Generations
Second Generation (1950-1965)Third Generation (1965-1975)
First generation (1945-1955)
Fourth Generation (1970-2010)
4.5 Generation
Fifth Generation (1995 - 2025) Aircraft Avionics
Third Generation Avionics
Fourth Generation Avionics
4.5 Generation AvionicsFifth Generation Avionics
Cockpit Displays
Communication (internal and external)Data Entry and Control
Flight Control
Fighter Aircraft Avionics I
Table of Content (continue – 1)
SOLO Fighter Aircraft Avionics
Aircraft Propulsion System
Aircraft Flight Performance
Navigation
Earth Atmosphere
Flight Instruments
Power Generation SystemEnvironmental Control System
Aircraft Aerodynamics
Fuel System
Jet Engine
Vertical/Short Take-Off and Landing (VSTOL)
Engine Control System
Flight Management System
Aircraft Flight Control
Aircraft Flight Control Surfaces
Aircraft Flight Control Examples
Fighter
Aircraft
Avionics
II
Table of Content (continue – 2)
SOLO
4
Fighter Aircraft Avionics
Equations of Motion of an Air Vehicle in Ellipsoidal Earth Atmosphere
Fighter Aircraft Weapon System
References
Safety Procedures
Tracking Systems
Aircraft Sensors
Airborne Radars
Infrared/Optical Systems
Electronic Warfare
Air-to-Ground Missions
BombsAir-to-Surface Missiles (ASM) or Air-to-Ground Missiles (AGM)
Fighter Aircraft Weapon Examples
Air-to-Air Missiles (AAM)
Fighter Gun
Avionics III
Continue fromFighter Aircraft Avionics
Part III
SOLO
5
Fighter Aircraft Avionics
SOLO
6
Fighter Aircraft Weapon System
AirDataSystem
Multi-Function Display
FlightInstrumentSystem
NavigationSystem
FlightManagementSystem
FlightControlSystem
AutopilotSystem
WeaponSystem
HUD HMD
AVIONICS DATA BUS
Infrared/OpticSensors
RadarSelf-Defense
System
SOLO
7
Fighter Aircraft Weapon System
Fighter/attack aircraft can carry a number of items fastened to racks underneath the aircraft.These items are called ‘‘Stores’’ and include Weapons (Bombs, Rockets, Missiles), Extra FuelTanks, Extra Sensor Pods, or Decoys (e.g., Chaff to fool radar-guided missiles and Flares tofool infra-red guided missiles). The Stores Management System (SMS) manages themechanical and electrical connections to weapons and senses their status under control ofthe Mission Central Computer (MCC); thus all weapons are readied via the SMS.
Weapons carried may include Rockets, Bombs (both Ballistic-dumb and Radar, Infra-red, or TVguided), and Missiles (which are typically ‘‘Fire and Forget’’ Self-guided using TV video, Laser,Imaging Infra-red, or Radar Seekers). Most aircraft also have internal fuselage-mountedGuns.
Weapon release modes include automatic (AUTO) and Continuously Computed Impact Point(CCIP) plus special modes for Guided Weapons. In AUTO mode, the MCC controls weaponrelease based on computed impact point, current target position, and predicted aircraft positionat release. In CCIP mode, the MCC computes a predicted impact point which is displayedon the HUD, and the aircrew controls weapon release with the bomb button on theHOTAS.
Stores
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8
Fighter Aircraft Weapon System
The Aircraft part of the Weapons System is checked for Operability and Safety on the Ground before the Weapons are Loaded. After the Weapons are Loaded on the Stations and Power (External or Aircraft Internal) and recognized in the Weapon System Inventory (Weapon Type and Station) the Weapons Power Bit check the Weapon Servicibility. This information is displayed to the Avionics.
The Weapons can be loaded on a Fighter Aircraft on the existing External Weapon Stations or if available on Internal Bay Stations (F-22, F-35) .
When the Aircraft is on the Ground the WeaponLaunching Signal are disabled. In addition, usually the Weapons are in a Safe Mode.
The Weapons can be Launched only when theAircraft is on the Air and the Pilot activated the MASTER ARM switch. The Launching sequence can Start after activated the Launch Switch that is usually located on the Flight Control Stick. The Launching sequence is defined to assure the Safety of the Launching Aircraft.
The Weapons System will indicate a Successful or Unsuccessful Launch and will choose the Next Weapon to be Launched according to a predefined sequence.Weapon Management Displays
SOLO Fighter Aircraft Weapon System
The Weapon System advises the Pilot how to Launch the Weapons.In general from the Third Fighter Generation and up the Aircraft Weapon System included a Computer that provided Flight Instruction Displays for the Pilot, to Release Bombs or Launch Missiles (A/A or A/G).
Target Designation
The Aircrew may designate a Target for A/A or A/G Attack in one of two ways: by Radar or by HUD/HMD designate. To designate a target by Radar, the Radar must already be tracking a Target. The Radar Target is identified as the Target by a Member of the Aircrew pushing the designate switch on the HOTAS. To perform a HUD/HMD designation, the Aircrew must first position the HUD/HMD reticle (on the HUD) using the Target Designator Controller (TDC) Switch on the HOTAS (the TDC Switch is similar to a Joystick). Once the HUD Reticle is properly positioned, the aircrew pushes down on the TDC switch to designate a target. The MCC must transform the HUD/HMD Reticle position from HUD coordinates to obtain Range, Azimuth, and Elevation to Target. No matter how the Target was designated, the HUD/HMD Reticle changes shape to indicate that a Target is Designated. A Designated Target may be undesignated by pushing the Undesignate Switch on the HOTAS.
SOLO Fighter Aircraft Weapon System
A/G Weapon Selection
Weapon selection includes selecting the type of Weapon, the number to drop, and thedesired spacing on the ground. This is done by the aircrew using the MPD stores display and Keyset switches. Depending on the type of weapon selected, a default delivery mode is defined and displayed. At any time prior to weapon release, the aircrew may push the AUTO/CCIP toggle switch on the Keyset, causing the delivery mode to change from AUTO to CCIP or from CCIP to AUTO. Weapon-ready determination is also assumed to be part of this function.
Mode Selection
The Pilot may choose between Air-to-Air (A/A) and Air-to-Ground (A/G)
Steering in A/G Mode
Compute the Steering Cues for display on the HUD/HMD and MPD based either on WaypointSteering or Target Attack Steering. The MCC can hold a Number of Aircrew-entered Waypoints (Latitude, Longitude, Altitude) which may be used as Steer-to Points and as Target Designation Points. The Aircrew may also associate an Offset (Range, Bearing) from the currently selected Waypoint which is taken into account. Prior to Target Designation, Steering Cues are provided based on the Currently Selected Waypoint (if any). After Target Designation, Steering Cues are provided based on Target Location relative to Aircraft Position
SOLO
Air-to-Ground Missions
11
Fighter Aircraft Weapon System
MULTI-COMMAND HANDBOOK 11-F16
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12
Fighter Aircraft Weapon System
Bombs:
-Dumb (Gravity) Bombs - Guided (Smart) Bombs * TV Bombs (Wallay) * Laser Guided Bombs (Paveway) * Gliding Bombs with Data Link and IR/Optical Seeker * Inertial/GPS Bombs (JDAM) * Inertial/GPS/EO (Spice) * Small Diameter Bombs
USAF artist rendering of JDAM kits fitted to Mk 84, BLU-109, Mk 83, and Mk 82 unguided bombs
GBU-39 Small Diameter Bomb
Armement Air-Sol Modulaire (Air-to-Ground Modular Weapon)
(AASM)
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13
Fighter Aircraft Weapon System
Dumb Bombs Delivery
There is the possibility to program visual cues in the computer of the F-16. Beside waypoints there are 4 types of cues. These are called VIP, VRP, PUP and OA’s. VIP = Visual Initial Point VRP = Visual Reference Point PUP = Pull Up Point OA = Offset Aim
The Bomb Delivery in Type 3 Fighters and up is done by the Weapon Delivery Computer.The Pilot chooses the Bomb Delivery Mode (TOSS, LAT, CCIP,..) in A/G Mode, Designates the Ground Target using the Gun Sight or HUD and after this the WeaponSystem provides Flight Instruction and Automatically Releases the Bombs.
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14
Fighter Aircraft Weapon System
Dumb Bombs Delivery (continue – 1)
Pop-Up This type of delivery can be useful for all static targets. Think about buildings, bridges, runways and even vehicles. The ordnance that can be used is the whole range from low and high drag dumb bombs, cluster and laser guided bombs.
TOSS (English word for throwing something up in the air)For a low level ingress we should use a LAT delivery. LAT stands for Low Altitude TOSS. During this delivery the bomb will be released upwards. The range will become greater but the accuracy smaller. Therefore the best type of bomb used will be a cluster bomb. This is a very nice way to attack a group of vehicles like a SA-2 or SA-3 site. But also freefall bombs can be used against large targets.
High Altitude Dive Bombing (HADB) This delivery should keep the attacker above a planned altitude and can be used for hitting all types of static target like buildings, bridges and vehicles. Any type of bomb can be used. It is also possible to use missiles like the AGM-65 with this delivery.
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15
Fighter Aircraft Weapon System
Dumb Bombs Delivery (continue – 2)
CCIP (Continuous Computed Impact Point)The objective of a CCIP delivery is to fly the Aircraft in a manner to arrive at or close to the Planned Release Parameters (Altitude, Airspeed and Dive Angle) with the CCIP Cue close to the Intended Aiming Point. When the CCIP Cue superimposes the Target, the Pickle Button / Trigger should be actuated to initiate Weapons Release / Firing
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16
Fighter Aircraft Weapon System
Dumb Bombs Delivery (continue – 3)
For Dumb Bombs the MCC solves the ballistic trajectory equations of motion.This is done initially to determine Weapon Time of Fall when the Estimated Time-to-Go toRelease (based on Aircraft Ground Speed and Target Ground Range) is less than one minute.Initialization must be repeated if a New Target is Designated. Once initialized, the WeaponTrajectory must be computed at least every 100 ms. Outputs include Time-to-Go to Release,Weapon Time of Fall, Down Range Error, and Cross Range Error. When Time-to-Go to Release falls below ΔT ms. and AUTO delivery mode is selected, Weapon Release is scheduled.Thereafter, whenever Time-to-Go to Release is recomputed, Weapon Release is rescheduled.
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17
Fighter Aircraft Weapon System
Air-to-Surface Missiles (ASM) or Air-to-Ground Missiles (AGM)
An air-to-surface missile (ASM) or air-to-ground missile (AGM or ATGM) is a missile designed to be launched from military aircraft (bombers, attack aircraft, fighter aircraft or other kinds) and strike ground targets on land, at sea, or both. They are similar to guided glide bombs but to be deemed a missile, they usually contain some kind of propulsion system. The two most common propulsion systems for air-to-surface missiles are Rocket Motors and Jet Engines. These also tend to correspond to the range of the missiles — short and long, respectively. Some Soviet air-to-surface missiles are powered by Ramjets, giving them both long range and high speed.
AGM-65 Maverick
Electro-optical, Laser, or Infra-red Guidance Systems
TAURUS KEPD 350
IBN (Image Based Navigation), INS (Inertial Navigation System), TRN (Terrain Referenced Navigation) and MIL-GPSGuidance System
Storm Shadow
Inertial, GPS and TERPROM. Terminal guidance using imaging infrared
AGM-158 JASSM (Joint Air-to-Surface Standoff Missile)
INS/GPS Guidance
18
An air-to-air missile (AAM) is a missile fired from an aircraft for the purpose of destroying another aircraft. AAMs are typically powered by one or more rocket motors, usually solid fuelled but sometimes liquid fuelled. Ramjet engines, as used on the MBDA Meteor (currently in development), are emerging as propulsion that will enable future medium-range missiles to maintain higher average speed across their engagement envelope.
Air-to-air missiles are broadly put in two groups. The first consists of missiles designed to engage opposing aircraft at ranges of less than approximately 20 miles (32 km), these are known as short-range or “within visual range” missiles (SRAAMs or WVRAAMs) and are sometimes called “dogfight” missiles because they emphasize agility rather than range. These usually use infrared guidance, and are hence also called heat-seeking missiles. The second group consists of medium- or long-range missiles (MRAAMs or LRAAMs), which both fall under the category of beyond visual range missiles (BVRAAMs). BVR missiles tend to rely upon some sort of radar guidance, of which there are many forms, modern ones also using inertial guidance and/or "mid-course updates".
Air-to-Air Missiles (AAM)
SOLO Fighter Aircraft Weapon System
A detailed description on the subject can be founded in the Power Point“Air Combat” Presentation. Here we give a brief summary of the subject.
Air- to-Air missile launch envelope
Kinematics no-escape-zone
Return to Table of Content
01-21
Air-to-Air Missiles Modes of OperationAir-to-Air Missiles Modes of Operation
Lock-On Before Launch
•High agility
•Tight radius turn
•Excellent minimum ranges
Active Homing Phase
• IMU alignment
• Radar slave- full target data
• HMD Slave- partial target data
• Seeker activation
• Target Lock-On
Pre Launch Phase
01-23
2
• Inertial navigation
• Trajectory shaping for maximum range
Midcourse Guidance Phase
• IMU alignment
• Target data transfer
Lock-On After Launch
3
• Seeker activation• Target Lock-On• Final homing
Homing Phase
1 Pre Launch Phase
AMRAAM
A/A MISSILES
AMRAAM AIM - 120C-5 SpecificationsLength: 12 ft 3.65 mDiameter: 7 in 17.8 cmWing Span: 17.5 in 44.5 cmFin Span: 17.6 in 44.7 cWeight: 356 lb 161.5 kgWarhead: 45 lb 20.5 KgGuidance: Active RadarFuzing: Proximity (RF) and ContactLauncher: Rail and eject
AIM-120CRocket motor PN G672798-1 is an enhanced version with additional 5” (12 cm) of propellant.Estimation: add 10% (12/140) to obtainmp ~ 52 kgWtot ~ 120,000 N s
AMRAAM AIM-120 Movie
Return to Table of Content
AIM-9X AIM-9X Movie
29
A-A Missiles Development in RAFAELA-A Missiles Development in RAFAEL
BVRBVR
Short RangeShort Range
PYTHON-4PYTHON-4
PYTHON-3PYTHON-3
SHAFRIR-2SHAFRIR-2
SHAFRIR-1SHAFRIR-1
PYTHON-5PYTHON-5
DERBYDERBY
Return to Table of Content
Rafael Python 5 Promo, Movie
Derby - Beyond Visual Range Air-to-Air Missile, Movie
30
Evolution of Air-to-Air Missiles in RAFAELEvolution of Air-to-Air Missiles in RAFAEL
PYTHON-4PYTHON-4
1st GENERATION
SHAFRIR-1SHAFRIR-1
2nd GENERATION
SHAFRIR-2SHAFRIR-2
3rd GENERATION
PYTHON-3PYTHON-3
4th GENERATION
SERVICE: SINCE 1993SERVICE SINCE 1978HITS: OVER 35 A/CDURING 1982 WAR
SERVICE: 1968-1980HITS: OVER 100 A/C
DURING 1973 WAR
SERVICE: 1964-1969
0-(10)
30
180
45
30
LEAD/LAGANGLE
0
MAX.ASPECT
ANGLE
TYPICAL 3rd
GENERATIONMISSILE
LAUNCHER
Short Range
DERBYDERBY
ACTIVR BVR
Dual Range
PYTHON-5
5th GENERATION
Full Sphere IR Missile
Full Scale Development
2.9
3. 6
Russian Air-to-Air Missiles
RVV-MD, RVV-BD New Generation Russian Air-to-Air Missiles, Movie
Russian Air Power, Movie
Russian Air Force vs USAF (NATO) Comparison, Movie
SU-30SM Intercept with R-77 Missile, Movie
Ukranian A-A Missile ALAMO, R-27, Movie
Return to Table of ContentReturn to Movies Table
People’s Republic of China (PRC) Air-to-Air Missiles
• PL - 1 - PRC version of the Soviet Kaliningrad K-5 (AA-1 Alkali), retired.
• PL - 2 - PRC version of the Soviet Vympel K-13 (AA-2 Atoll), based on AIM-9 Sidewinder, retired.
• PL - 3 - updated version of the PL-2, did not enter service. PL-2, 3
• PL - 5 - updated version of the PL-2, several versions:
• PL - 5A - Semi-Active Radar homing AAM, resembles AIM-9G. Did not enter service
• PL - 5B - IR version, entered service 1990 to replace PL-2. Limited of boresight.
• PL - 5C - Improved version comparable to AIM-9H or AIM-9L in performance.
• PL - 5E - All-aspect attack version, resembles AIM-9P in appearance.
• PL - 7 - PRC version of the IR-homing French R550 Magic AAM. Did not enter service.• PL - 8 - PRC version of the Israeli RAFAEL Python 3.• PL - 9 - short range IR missile, marked for export. One known improved version PL - 9C.
• PL - 10 - medium-range air-to-air missile. Did not enter service.
PL-5
PL-8
PL-9
PL-7
People’s Republic of China (PRC) Air-to-Air Missiles (continue)
• PL - 11 - Medium Range Air-to-Air Missile (MRAAM), based on the HQ-61C and Italian ASPIDE (AIM-7)technology. Known version include:
PL -11Length: 3.690 mBody diameter: 200 mmWing span: 1 mLaunch weight: 220 kgWarhead: HE-fragmentationFuze: RFGuidance: Semi-Active CW RadarPropulsion: Solid propellantRange: 25 km
• PL - 11 - MRAAM with semi-active radar homing, based on the HQ-61C SAM and ASPIDE seeker technology. Exported as FD-60.
• PL - 11A - Improved PL-11 with increased range, warhead, and moreeffective seeker. The new seeker requires target illuminationonly during the last stage, providing a Lock On After Launchcapability.
• PL - 11B - Also known as PL-11AMR, improved PL-11 with AMR-1,active radar-homing seeker.
• LY - 60 - PL-11, adopted to navy ships for air-defense, sold to Pakistanbut doesn’t appear to be in service with the Chinese Navy.
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34
Fighter Aircraft Weapon System
F4-Phantom Armament
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35
Fighter Aircraft Weapon System
F-16
SOLO
36
Fighter Aircraft Weapon System
http://www.freerepublic.com/focus/f-news/2845813/posts
F-15
SOLO
37
Fighter Aircraft Weapon SystemF-15C: M61A1 Vulcan Cannon and AIM-9M Sidewinder, Movie
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38
Fighter Aircraft Weapon System
F-18
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39
Fighter Aircraft Weapon System
The F/A-18 E/F Super Hornet, with its array of weapons systems, is the world's most advanced high-performance strike fighter. Designed to operate from aircraft carriers and land bases, the versatile Super Hornet can undertake virtually any combat mission.
SOLO
F-22 Raptor
http://www.ausairpower.net/APA-Raptor.html
Fighter Aircraft Weapon System
40
Internal WeaponBay
41
Lockheed_Martin_F-35_Lightning_II
Fifth Generation Avionics
F-35 Simulator - AA and AG Modes _ Avionics-1, Movie
Lockheed_Martin_F-35_Lightning_II
Fifth Generation Avionics
42
43
Fighter Aircraft Weapon System
Su-32/34
44
Fighter Aircraft Weapon System
45
Su-35
Fighter Aircraft Weapon System
46
Fighter Aircraft Weapon System
47
Fighter Aircraft Weapon System
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48
Fighter Aircraft Weapon System
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49
Fighter Aircraft Weapon System
SOLO
50
Fighter Aircraft Weapon System
SOLO
51
Fighter Aircraft Weapon System
Fighter Gun
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52
Fighter Aircraft Weapon System
53
Performance of Aircraft Cannons in terms of their Employment in Air Combat
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54
Performance of Aircraft Cannons in terms of their Employment in Air CombatSOLO
55
Performance of Aircraft Cannons in terms of their Employment in Air Combat
SOLO
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56
Safety Procedures
Safety of Personal when the Aircraft is on the Ground and when it is in the Air.Avionics includes Safety Procedures:
Fighter Aircraft on the GroundIn this case the Aircraft Weight is sustained by the Wheels and a Weight-on-WheelsSwitch (WOW) and the Master Arm (MA) Switch are in Safe Mode preventing the Release/Fire Signals to reach the Weapon Storage
Ground Crew will perform the following: * Visual Check of the Unpowered Aircraft * Connect an External Power Generator and will check the Avionics Serviceability * By pressing WOW Safety-Override and MA=ARM will check the Weapon Release System. * Disconnect the External Power Generator and Load the Weapons on Storage * Install the Weapons External Safety Devices, to be removed before Taxiing to Take Off. In general, the Weapons have also internal Safety Devices. * Reconnect External Power Generator, insert the Weapons in the SMS Inventory, (WOW = Safe) and perform Power On BIT of the Weapons to check their Serviceability. * Disconnect the External Power Generator and the Aircraft (already fueled) is ready to be delivered to the Air Crew.
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57
Safety Procedures
Safety of Personal when the Aircraft is on the Ground and when it is in the Air.Avionics includes Safety Procedures (continue – 1):
Fighter Aircraft on the GroundIn this case the Aircraft Weight is sustained by the Wheels and a Weight-on-WheelsSwitch (WOW) and the Master Arm (MA) Switch are in Safe Mode preventing the Release/Fire Signals to reach the Weapon Storage
Air Crew will perform the following: * Visual Check of the Unpowered Aircraft * Start the Engines that provide Internal Power and will check the Avionics Serviceability (WOW = Safe and MA = Safe) * Insert the Weapons in the SMS Inventory, and perform Power On BIT of the Weapons to check their Serviceability. * Input to Avionics Data necessary for the Mission. * The Avionics will be in NAV Mode. * Before Taxiing to Take Off the Ground Crew will remove all Weapons Safety Devices. * Pilot will Taxi and Take Off. * After Landing the Ground Crew will Reinstall Weapons Safety Devices.
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58
Safety Procedures
Safety of Personal when the Aircraft is on the Ground and when it is in the Air.Avionics includes Safety Procedures (continue – 2):
Fighter Aircraft in the AirIn this case the Weight-on-Wheels Switch (WOW) is in ARM.MA = Safe preventing Release/Launch of Weapons.To operate the Weapons the pilot must put MA = Arm.The Pilot can switch between the three Operational Modes: - NAV : Navigation Mode - A/A: can Launch A/A Missiles and Fire Gun Projectiles - A/G: can Launch A/G Missile or release Bombs
The Avionics will deliver Safety Warnings due to - An Aircraft Malfunction - A Flight Hazard - Fuel Shortage
In case of a Weapon Release Malfunction the Pilot may:• Jettison the Weapon• Perform Safety Procedures at Landing.
59
SOLO AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
1. Inertial System Frame
2. Earth-Center Fixed Coordinate System (E)
3. Earth Fixed Coordinate System (E0)
4. Local-Level-Local-North (L) for a Spherical Earth Model
5. Body Coordinates (B)
6. Wind Coordinates (W)
7. Forces Acting on the Vehicle
8. Simulation
8.1 Summary of the Equation of Motion of a Variable MassSystem
8.2 Missile Kinematics Model 1 (Spherical Earth)
8.3 Missile Kinematics Model 2 (Spherical Earth)
60
Bz
MV
Bx
ByWy
WzBr
Bp
Wp
BqWqWr
Given a missile with a jet engine, we define:
1. Inertial System Frame III zyx ,,
3. Body Coordinates (B) , with the origin at the center of mass. BBB zyx ,,
2. Local-Level-Local-North (L) for a Spherical Earth Model LLL zyx ,,
4. Wind Coordinates (W) , with the origin at the center of mass. WWW zyx ,,
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERESOLO
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
Coordinate Systems
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SOLO
Coordinate Systems
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
1 .Inertial System (I)
R
- vehicle position vector
Itd
RdV
- vehicle velocity vector, relative to inertia
IItd
Rd
td
Vda
2
2
- vehicle acceleration vector, relative to inertia
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
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SOLO
Coordinate Systems (continue – 2)
2. Earth Center Fixed Coordinate System (E) xE, yE in the equatorial plan with xE pointed to the intersection between the equatorto zero longitude meridian.
The Earth rotates relative to Inertial system I, with the angular velocity
sec/10.292116557.7 5 rad
EIIE zz
11
0
0EC
IE
Rotation Matrix from I to E
100
0cossin
0sincos
3 tt
tt
tC EI
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
63
SOLO
Coordinate Systems (continue – 3)
2 .Earth Fixed Coordinate System (E) (continue – 1)
Vehicle Position ETEI
EIE
I RCRCR
Vehicle Velocity
Vehicle Acceleration
RVRtd
Rd
td
RdV EIE
EI
- vehicle velocity relative to Inertia
Rtd
Rd
td
RdV IE
LE
E
: - vehicle velocity relative to Earth
II
E
I
E
I
Rtd
d
td
VdRV
td
d
td
Vda
RVtd
VdR
td
RdR
td
dV
td
VdEIEEU
U
E
EE
EIU
U
E
IU
0
RVtd
VdRV
td
Vda E
E
EEEU
U
E
22
or
where U is any coordinate system. In our case U = E.
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
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SOLO
Coordinate Systems (continue – 4)
3 .Earth Fixed Coordinate System (E0)
The origin of the system is fixed on the earth at somegiven point on the Earth surface (topocentric) of Longitude Long0 and latitude Lat0.
xE0 is pointed to the geodesic North, yE0 is pointed to the East parallel to Earthsurface, zE0 is pointed down.
100
0cossin
0sincos
sin0cos
010
cos0sin
2/ 00
00
00
00
30200 LongLong
LongLong
LatLat
LatLat
LongLatC EE
00000
00
00000
sinsincoscoscos
0cossin
cossinsincossin
LatLongLatLongLat
LongLong
LatLongLatLongLat
The Angular Velocity of E relative to I is: EIIEIE zz
110 or
0
0
00000
00
00000
000
sin
0
cos
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
Lat
Lat
LatLongLatLongLat
LongLong
LatLongLatLongLat
C EE
EIE
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
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SOLO
Coordinate Systems (continue – 5)
4 .Local-Level-Local-North (L) The origin of the LLLN coordinate system is located atthe projection of the center of gravity CG of the vehicleon the Earth surface, with zDown axis pointed down, xNorth, yEast plan parallel to the local level, withxNorth pointed to the local North and yEast pointed tothe local East. The vehicle is located at:.
Latitude = Lat, Longitude = Long, Height = H
Rotation Matrix from E to L
100
0cossin
0sincos
sin0cos
010
cos0sin
2/ 32 LongLong
LongLong
LatLat
LatLat
LongLatC LE
LatLongLatLongLat
LongLong
LatLongLatLongLat
sinsincoscoscos
0cossin
cossinsincossin
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
66
SOLO
Coordinate Systems (continue – 6)
4 .Local-Level-Local-North (L) (continue – 1)
Angular Velocity
IEELIL Angular Velocity of L relative to I
Lat
Lat
LatLongLatLongLat
LongLong
LatLongLatLongLat
C LE
Down
East
NorthL
IE
sin
0
cos
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
LatLong
Lat
LatLong
Lat
LongLatLongLatLongLat
LongLong
LatLongLatLongLat
Lat
Long
C LE
Down
East
NorthL
EL
sin
cos
0
0
0
0
sinsincoscoscos
0cossin
cossinsincossin
0
0
0
0
LatLong
Lat
LatLong
DownDown
EastEast
NorthNorthL
IECL
ECLL
IL
sin
cos
Therefore
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
67
SOLO
Coordinate Systems (continue – 7)
4 .Local-Level-Local-North (L) (continue – 2)
Vehicle Velocity
Vehicle Velocity relative to I
RVRtd
Rd
td
RdV EIE
EI
HRLatLongLat
LatLongLatLong
LatLatLong
HR
Rtd
RdV EL
L
LE
00
0
0
0cos
cos0sin
sin0
0
0
where is the vehicle velocity relative to Earth.EV
DownE
EastE
NorthE
V
V
V
H
HRLatLong
HRLat
_
_
_
0
0
cos
from which
DownE
EastE
NorthE
Vtd
Hd
LatHR
V
td
Longd
HR
V
td
Latd
_
0
_
0
_
cos
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
HeightVehicleHRadiusEarthmRHRR 600 10378135.6
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
68
SOLO
Coordinate Systems (continue – 8)
4 .Local-Level-Local-North (L) (continue – 3)
Vehicle Velocity (continue – 1)
We assume that the atmosphere movement (velocity and acceleration) relative to EarthAt the vehicle position (Lat, Long, H) is known. Since the aerodynamic forces on thevehicle are due to vehicle movement relative to atmosphere, let divide the vehiclevelocity in two parts:
WAE VVV
Down
East
NorthL
A
V
V
V
V
- Vehicle Velocity relative to atmosphere
DownW
EastW
NorthW
LW
V
V
V
HLongLatV
_
_
_
,,
- Wind Velocity at vehicle position (known function of time)
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
69
SOLO
Coordinate Systems (continue – 9)
4 .Local-Level-Local-North (L) (continue – 4)
Vehicle Acceleration
Since:
RVtd
VdR
td
d
td
VdRV
td
d
td
Vda EEL
L
E
II
E
I
E
I
2
WAE VVV
WWIL
L
WAAIL
L
A VVtd
VdRVV
td
Vda
Wa
WWEL
L
WAAEL
L
A VVtd
VdRVV
td
Vd 22
HLongLatVHLongLattd
VdHLongLata WEL
L
WW ,,2,,:,,
WAAEL
L
A aRVVtd
Vd
2
where:
is the wind acceleration at vehicle position.
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
Table of Content
70
SOLO
Coordinate Systems (continue – 10)
5 .Body Coordinates (B)
The origin of the Body coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xB pointedto the front of the Air Vehicle, yB pointedtoward the right wing and zB completingthe right-handed Cartesian reference frame.
Bx
Lx
Bz
Ly
LzBy
Rotation Matrix from LLLN to B (Euler Angles):
cccssscsscsc
csccssssccss
ssccc
C BL 321
- azimuth angle
- pitch angle
- roll angle
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
71
SOLO
Coordinate Systems (continue – 11)
5 .Body Coordinates (B) (continue – 1)
Bx
Lx
Bz
Ly
LzBy
Angular Velocity from L to B (Euler Angles):
0
0
0
0
0
0 211
R
Q
PB
LB
0
0
cos0sin
010
sin0cos
cossin0
sincos0
001
0
0
cossin0
sincos0
001
0
0
G
coscossin0
cossincos0
sin01
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
72
SOLO
Coordinate Systems (continue – 12)
5 .Body Coordinates (B) (continue – 2)
Bx
Lx
Bz
Ly
LzBy
Rotation Matrix from LLLN to B (Quaternions):
321
412
143
234
3412
2143
1234
44 3333
BIBLBL
BLBLBL
BLBLBL
BLBLBL
BLBLBLBL
BLBLBIBL
BLBLBLBL
TBLBLBLXBLBLXBL
BL
qqq
qqq
qqq
qqq
qqqq
qqqq
qqqq
qqqIqqIqC
where:
3
2
1
:&4
4
3
2
1
4
3
2
1
BL
BL
BL
BLBL
BLBL
BL
BL
BL
BL
BL
BL
BL
BL
BL
q
q
q
qqor
q
q
q
q
q
q
q
q
q
2sin
2sin
2sin
2cos
2cos
2cos4
BLq
2cos
2sin
2sin
2sin
2cos
2cos1
BLq
2sin
2cos
2sin
2cos
2sin
2cos2
BLq
2sin
2sin
2cos
2cos
2cos
2sin3
BLq
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
73
SOLO
Coordinate Systems (continue – 13)
5 .Body Coordinates (B) (continue – 3)
Bx
Lx
Bz
Ly
LzBy
Rotation Matrix from LLLN to B (Quaternions))continue – 1(
The quaternions are given by the followingdifferential equations:
BL
LIL
BIBBLBLBL
BILBL
BIBBL
BIL
BIBBL
BLBBLBL qqqqqqqqq
2
1
2
1*
2
1
2
1
2
1
2
1
04321
3412
2143
1234
2
1
4
3
2
1
B
B
B
BLBLBLBL
BLBLBLBL
BLBLBLBL
BLBLBLBL
BL
BL
BL
BL
r
q
p
qqqq
qqqq
qqqq
qqqq
q
q
q
q
4
3
2
1
0
0
0
0
2
1
BL
BL
BL
BL
zLzLyLyLxLxL
zLzLxLxLyLyL
yLyLxLxLzLzL
xLxLyLyLzLzL
q
q
q
q
4
3
2
1
0
0
0
0
2
1
BL
BL
BL
BL
zLzLByLyLBxLxLB
zLzLBxLxLByLyLB
yLyLBxLxLBzLzLB
xLxLByLyLBzLzLB
q
q
q
q
rqp
rpq
qpr
pqr
or:
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
74
SOLO
Coordinate Systems (continue – 14)
5 .Body Coordinates (B) (continue – 4)
Bx
Lx
Bz
Ly
LzBy
Vehicle Velocity
Vehicle Velocity relative to Earth is divided in:
WAE VVV
w
v
u
V BA
DownW
EastW
NorthW
BL
zW
yW
xW
BW
V
V
V
C
V
V
V
HLongLatV
B
B
B
_
_
_
,,
Vehicle Acceleration
WWIB
B
WAAIB
B
A
I
VVtd
VdRVV
td
Vd
td
Vda
W
AELALB
B
A
a
RVVtd
Vd
2
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Table of Content
75
SOLO
Coordinate Systems (continue – 15)
6 .Wind Coordinates (W)
Bx
Lx
Bz
Ly
LzBy
Wz
V
The origin of the Wind coordinate systemis located at the instantaneous center ofgravity CG of the vehicle, with xW pointedin the direction of the vehicle velocity vectorrelative to air .AV
cos0sin
sinsincossincos
cossinsincoscos
cos0sin
010
sin0cos
100
0cossin
0sincos
23WBC
The Wind coordinate frame is defined by the following two angles:
- angle of attack
- sideslip angle
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
76
SOLO
Coordinate Systems (continue – 16)
6 .Wind Coordinates (W) (continue -1)
Bx
Lx
Bz
Ly
LzBy
Wz
V
Rotation Matrix from L (LLLN) to W is:
- azimuth angle of the trajectory
- pitch angle of the trajectory
Rotation Matrix
32123 BL
WB
WL CCC
The Rotation Matrix from L (LLLN) to W can also be defined by the following Consecutive rotations:
- bank angle of the trajectory
cccssscsscsc
csccssssccss
ssccc
CC WL
WL 321
*1
We defined also the intermediate wind frame W* by:
csscs
cs
ssccc
CWL 032
*
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
77
SOLO
Coordinate Systems (continue – 17)
6 .Wind Coordinates (W) (continue -2)
Bx
Lx
Bz
Ly
LzBy
Wz
V
Angular Velocity of W* relative to LLLN is:
Angular Velocities
cos
sin
0
0
cos0sin
010
sin0cos
0
0
0
0
0
0
2
*
*
**
*
W
W
WW
LW
R
Q
P
Angular Velocity of W relative to LLLN is:
coscossin0
cossincos0
sin01
cos
sin
cossin0
sincos0
001
0
00
0
0
0
0
0 21
W
W
WW
LW
R
Q
P
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
78
SOLO
Coordinate Systems (continue – 18)
6 .Wind Coordinates (W) (continue -3)
Bx
Lx
Bz
Ly
LzBy
Wz
V
We have also:
Angular Velocities (continue – 1)
Down
East
North
WL
WL
LIE
WL
zW
yW
xWW
IE C
Lat
Lat
CC ***
*
*
**
sin
0
cos
Down
East
North
WL
WL
LEL
WL
zW
yW
xWW
EL C
LatLong
Lat
LatLong
CC
***
*
*
**
sin
cos
*
1
sin
0
cosW
IEWL
LIE
WL
zW
yW
xWW
IE
Lat
Lat
CC
*1
sin
cos
WIL
WL
LIL
WL
WIL
LatLong
Lat
LatLong
CC
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
79
SOLO
Coordinate Systems (continue – 19)
6 .Wind Coordinates (W) (continue -4)
Bx
Lx
Bz
Ly
LzBy
Wz
V
The Angular Velocity from I to W is:
Angular Velocities (continue – 2)
DownDown
EastEast
NorthNorth
WL
W
W
WL
ILWL
W
W
WW
ILW
LW
W
W
WW
IW C
R
Q
P
C
R
Q
P
r
q
p
Using the angle of attack α and the sideslip angle β , we can write:
BWBW yz
11
or:
0
0
0
0
3
r
q
p
C
r
q
pWB
W
W
WW
IBW
IWW
BW
but also:
0
0
0
0
3
R
Q
P
C
R
Q
PWB
W
W
WW
LBW
LWW
BW
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
80
SOLO
Coordinate Systems (continue – 20)
6 .Wind Coordinates (W) (continue -5)
Bx
Lx
Bz
Ly
LzBy
Wz
V
We can write:
Angular Velocities (continue – 3)
0
cos
sin
0
0
cos0sin
sinsincossincos
cossinsincoscos
r
q
p
r
q
p
W
W
W
or:
cossin
sinsincossincos
cossinsincoscos
rpr
rqpq
rqpp
W
W
W
This can be rewritten as:
tansincoscos
rpq
q W
Wrrp cossin
cos
sinsincos
tantansincossincossincossincos
W
WW
qrp
qrpqrpp
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
81
SOLO
Coordinate Systems (continue – 21)
6 .Wind Coordinates (W) (continue -6)
Bx
Lx
Bz
Ly
LzBy
Wz
V
We have also:
Angular Velocities (continue – 4)
tansincoscos
RPQ
Q W
WRRP cossin
cos
sinsincos WW
QRPP
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
82
SOLO
Coordinate Systems (continue – 22)
6 .Wind Coordinates (W) (continue -7)
Bx
Lx
Bz
Ly
LzBy
Wz
V
The vehicle velocity was decomposed in:
Vehicle Velocity
WAE VVV
0
0
V
V WA
- vehicle velocity relative to atmosphere
DownW
EastW
NorthW
WL
zW
yW
xW
WW
V
V
V
C
V
V
V
HLongLatV
W
W
W
_
_
_
,,
- wind velocity at velocity position
also
0
0
0
011*
VV
VV WA
WA
DownW
EastW
NorthW
WL
zW
yW
xW
WW
V
V
V
C
V
V
V
HLongLatV
W
W
W
_
_
_
*
*
*
*
* ,,
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
83
SOLO
Coordinate Systems (continue – 23)
6 .Wind Coordinates (W) (continue -8)
Bx
Lx
Bz
Ly
LzBy
Wz
V
The vehicle acceleration in W* coordinates is
Vehicle Acceleration
WAELALW
W
A
WWIW
W
WAAIW
W
A
I
C
aRVVtd
Vd
VVtd
VdRVV
td
Vd
td
Vda
2*
*
*
*
*
*
from which
*******
*
*
*
2 WW
WA
WWEL
WWA
WLW
W
W
A aVAVtd
Vd
where
RaA
:
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
84
SOLO
Coordinate Systems (continue – 24)
6 .Wind Coordinates (W) (continue -9)
Bx
Lx
Bz
Ly
LzBy
Wz
V
Vehicle Acceleration (continue – 1)
**
*
*
****
****
****
*
*
*
**
**
**
0
0
022
202
220
0
0
0
0
0
0
0
zWW
yWW
xWW
xWxWyWyW
xWxWzWzW
yWyWzWzW
zW
yW
xW
WW
WW
WW
a
a
aV
A
A
AV
PQ
PR
QRV
where
HR
Lat
Lat
C
a
a
a
A
A
A
A WL
zW
yW
xW
zW
yW
xW
W
2*
*
*
*
*
*
*
*
sin
0
cos - Acceleration due to external forces on the
Air Vehicle in W* coordinates
That gives
*****
*****
**
2
2
zWWyWyWzWW
yWWzWzWyWW
xWWxW
aVAVQ
aVAVR
aAV
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
85
SOLO
Coordinate Systems (continue – 25)
6 .Wind Coordinates (W) (continue -10)
Bx
Lx
Bz
Ly
LzBy
Wz
V
Vehicle Acceleration (continue – 2)
Using
cos
sin
*
*
**
*
W
W
WW
LW
R
Q
P
we have
** xWWxW aAV
cos/2 ****
zWzW
yWWyW
V
aA
****
2 yWyWzWWzW
V
aA
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Table of Content
86
SOLO
Aerodynamic Forces npp ˆt̂
n̂V
ds
wx1
wy1
wz1
tf ˆ
Pressure force
Friction force
WS
WS
A dstfnppF
11
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ
airflowingthebyweatedsurfaceVehicleS
SsurfacetheonmNstressforcefrictionf
Ssurfacetheondifferencepressurepp
W
W
W
)/( 2
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
7. Forces Acting on the Vehicle
87
SOLO
7. Forces Acting on the Vehicle (continue – 1)
Bx
Lx
Bz
Ly
LzBy
Wz
V
WyT
C
L
D
g
Aerodynamic Forces (continue – 1)
L
C
D
F WA
ForceLiftL
ForceSideC
ForceDragD
L
C
D
CSVL
CSVC
CSVD
2
2
2
2
12
12
1
tCoefficienLiftRMC
tCoefficienSideRMC
tCoefficienDragRMC
eL
eC
eD
,,,
,,,
,,,
ityvisdynamic
lengthsticcharacteril
soundofspeedHa
numberynoldslVR
numberMachaVM
e
cos
)(
Re/
/
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
88
SOLO
7. Forces Acting on the Vehicle (continue – 2)
Aerodynamic Forces (continue -2)
W
W
W
SfpL
SfpC
SfpD
dswztCwznCS
C
dswytCwynCS
C
dswxtCwxnCS
C
1ˆ1ˆ1
1ˆ1ˆ1
1ˆ1ˆ1
nCq p ˆt̂
n̂V
ds
wx1
wy1
wz1
tCq fˆ
Pressure force
Friction force
WS SVq 2
2
1
Wf
Wp
SsurfacetheontcoefficienfrictionV
fC
SsurfacetheontcoefficienpressureV
ppC
2/
2/
2
2
ntonormalplanonVofprojectiont
dstonormaln
ˆˆ
ˆ
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
89
MomentFriction
S
C
Momentessure
S
CCA
WW
dstRRfdsnRRppM 11
Pr
/
Aerodynamic Moments Relative to C can be divided in Pressure Moments andFriction Moments.
FrictionSkinorFrictionViscous
S
essureNormal
S
A
WW
dstfdsnppF 11
Pr
fp
V
ASALM
Aerodynamic Forces can be divided in Pressure Forces and Friction Forces.
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
npp ˆt̂
n̂V
ds
wx1
wy1
wz1
tf ˆ
Pressure force
Friction force
WS
AERODYNAMIC FORCES AND MOMENTS.
90
SOLO
iopenS
outflowoutopenflowinflowinopenflow dsnppmVmVT
1:
0
/
0
/ THRUST FORCES
iopenS
OoutflowoutopenflowCoutopeninflowinopenflowCiopenCT dsnppRRmVRRmVRRM
1:
0
/
0
/,
THRUST MOMENTS RELATIVE TO C
inopenS
inflowinopenflow dsnppmV
1
00
/
outopenS
outflowoutopenflow dsnppmV
1
0
/
T
outopenR
iopenR
CR
C
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Table of Content
CTM ,
91
SOLO
7. Forces Acting on the Vehicle (continue – 3)
BxLx
Bz
Ly
LzBy
Wz
V
Wy
T
C
L
D
gT
T
Thrust
B
B
B
z
y
x
BWB
W
T
T
T
TCT
cos0sin
sinsincossincos
cossinsincoscos**
*
*
*
cossin0
sincos0
001*
1
W
W
W
W
W
W
z
y
x
W
z
y
x
W
T
T
T
T
T
T
T
T
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Bx
Lx
Bz
Ly
LzBy
Wz
V
WyT
C
L
D
g
92
SOLO
7. Forces Acting on the Vehicle (continue – 4)
Gravitation Acceleration
zgygxg
gg100
0
0
0
010
0
0
0
001
cs
sc
cs
sc
cs
scC EWE
W
gg
cc
cs
sW
2sec/174.322sec/81.90
2
0
00gg ftmg
HR
R
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
The derivation of Gravitation Acceleration assumes an Ellipsoidal Symmetrical Earth.The Gravitational Potential U (R,ϕ) is given by
,
sin1,2
RUg
PR
aJ
RRU
EE
n n
n
n
μ – The Earth Gravitational Constanta – Mean Equatorial Radius of the EarthR=[xE
2+yE2+zE
2]]/2 is the magnitude of the Geocentric Position Vectorϕ – Geocentric Latitude (sinϕ=zE/R)Jn – Coefficients of Zonal Harmonics of the Earth Potential FunctionPn (sinϕ) – Associated Legendre Polynomials
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
93
SOLO
7. Forces Acting on the Vehicle (continue – 5)
Gravitation Acceleration
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Retaining only the first three terms of theGravitational Potential U (R,ϕ) we obtain:
R
z
R
z
R
z
R
aJ
R
z
R
aJ
Rg
R
y
R
z
R
z
R
aJ
R
z
R
aJ
Rg
R
x
R
z
R
z
R
aJ
R
z
R
aJ
Rg
EEEEz
EEEEy
EEEEx
E
E
E
342638
515
2
31
342638
515
2
31
342638
515
2
31
2
2
4
44
42
22
22
2
2
4
44
42
22
22
2
2
4
44
42
22
22
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
sin
cossin
coscos
R
zR
yR
x
E
E
E
2/1222EEE zyxR
94
SOLO
23. Local Level Local North (LLLN) Computations for an Ellipsoidal Earth Model
2
2210
20
20
20
5
21
20
60
sin
sin1
sin321
sin1
sec/10292116557.7
sec/051646.0
sec/780333.9
26.298/.1
10378135.6
Ae
e
p
m
e
HR
RLatggg
LateRR
LateeRR
LateRR
rad
mg
mg
e
mR
LatHR
V
HR
V
HR
V
Ap
EastDown
Am
NorthEast
Ap
EastNorth
tan
Lat
Lat
Down
East
North
sin
0
cos
DownDownDown
EastEast
NorthNorthNorth
East
North
Lat
LatLong
cos
t
t
dtLatLattLat
dtLongLongtLong
0
0
0
0
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
I
0Ex
0Ey
Iz
Northx
EastyDownz
Bx
ByBz
Iy
Ixt
tLong
Lat
0Ez
Ex
Ey
Ez
AV
SIMULATION EQUATIONS
95
SOLO
Down
East
North
Down
East
North
1 Latcos
1
LatHR
V
HR
V
HR
V
Ap
EastDown
Am
NorthEast
Ap
EastNorth
tan
Long
Lat
Down
East
North
sin
0
cos
Down
East
North
L
V
V
V
V
2
2210
20
20
20
5
21
20
60
sin
sin1
sin321
sin1
sec/10292116557.7
sec/051646.0
sec/780333.9
26.298/.1
10378135.6
Ae
e
p
m
e
HR
RLatggg
LateRR
LateeRR
LateRR
rad
mg
mg
e
mR
s
1
s
1
DownDownDown
EastEast
NorthNorthNorth
pR
mR
AH
Long
0Long
Lat
0Lat
Lat
Long
g
Lat g
LOCAL LEVEL LOCAL NORTHCOMPUTATIONS
Lat
North DownEast
AIR VEHICLE IN ELLIPTICAL EARTH ATMOSPHERESIMULATION EQUATIONS
Table of Content
96
SOLO
7. Forces Acting on the Vehicle (continue – 6)
Force Equations
Bx
Lx
Bz
Ly
LzBy
Wz
V
WyT
C
L
D
g
Air Vehicle Acceleration
WAELALW
W
A
I
C aRVVtd
Vd
td
Vda
2
WAELALW
W
AA aRVV
td
VdamTF
m
2g
1
Rg
g: where
ccgm
LT
csgm
CT
sgm
DT
A
A
A
zW
yW
xW
sin
sincos
coscos
cg
sg
m
LTm
CTm
DT
A
A
A
zW
yW
xW
0
sin
sincos
coscos
cossin0
sincos0
001
*
*
*
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
Table of Content
97
SOLO
LBL
BBA
BCG gCT
mF
ma
11
B
BrCrrotorB
IBB
BrCrrotor
BIBC
BIB
BIBCCTCAC
BIB
II
IIMMI
��
��
,,
,,,,1
BCG
TBL
LCG aCa
BIB
BL
LIL
BIBBLBL qqq
2
1
2
1
s
1
CT
CA
M
M
,
,
TBL IqIqC
3434
BIB
BCGa
L
CGa
B
BA
T
F
BLC
BLC
s
1 BLqBLq
BLC
s
1 L
ELL
EL
LLCG
LE VRaV
2 s
1 L
EV L
EV
LCGa
BLC
LMR
L
EV
LM
BL
BM VCV
Mee
22M
s
1
s
1
LEV
23 WBC
WBC
MV
WEM VVV
LMV
LWV
BIB
BBrotor
BBrotor
,
Missile Kinematics Model 1 in Vector Notation (Spherical Earth)
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
98
SOLO
rB
rB
r
rotor
B
B
B
zzyzxz
yzyyxy
xzxyxx
BB
BB
BB
B
B
B
zzyzxz
yzyyxy
xzxyxx
zBC
yBC
xBC
zBCA
yBCA
xBCA
zzyzxz
yzyyxy
xzxyxx
B
B
B
q
rI
r
q
p
III
III
III
pq
pr
qr
r
q
p
III
III
III
T
T
T
M
M
M
III
III
III
r
q
p
0
0
0
,
,
,
,
,
,
1
rr ,
B
B
B
r
q
p
s
1
B
B
B
r
q
p
B
B
B
r
q
p
4
3
2
1
0
0
0
0
2
1
4
3
2
1
BL
BL
BL
BL
zLzLByLyLBxLxLB
zLzLBxLxLByLyLB
yLyLBxLxLBzLzLB
xLxLByLyLBzLzLB
BL
BL
BL
BL
q
q
q
q
rqp
rpq
qpr
pqr
q
q
q
q
s
1
4
3
2
1
BL
BL
BL
BL
q
q
q
q
4
3
2
1
BL
BL
BL
BL
q
q
q
q
g
C
C
C
T
T
T
mF
F
F
ma
a
a
BL
BL
BL
zB
yB
xB
zBA
yBA
xBA
zB
yB
xB
3,3
3,2
3,111
zBzBA
yByBA
xBxBA
TF
TF
TF
zB
yB
xB
a
a
a
zB
yB
xB
BL
BL
BL
BL
BL
BL
BL
BL
BL
Down
East
North
a
a
a
CCC
CCC
CCC
a
a
a
3,33,23,1
2,32,22,1
1,31,21,1
Down
East
North
a
a
a
BLC
BLC
4
3
2
1
*
1
4
3
2
1
BL
BL
BL
BL
BL
BL
BL
BL
q
q
q
q
q
q
q
q 4
3
2
1
BL
BL
BL
BL
q
q
q
qB
LC
321
412
143
234
3412
2143
1234
BLBLBL
BLBIBL
BLBLBL
BLBLBL
BLBLBLBL
BLBLBLBL
BLBLBLBLB
L
qqq
qqq
qqq
qqq
qqqq
qqqq
qqqq
C
4
3
2
1
BL
BL
BL
BL
q
q
q
q
Down
East
North
a
a
a
DownW
EastW
NorthW
NorthNorthEastEast
NorthNorthDownDown
EastEastDownDown
Down
East
North
DownE
EastE
NorthE
V
V
V
Lat
Lat
HR
a
a
a
V
V
V
022
202
220
sin
0
cos2
_
_
_
s
1
DownE
EastE
NorthE
V
V
V
cos0sin
sinsincossincos
cossinsincoscosW
BC
WBC
s
1
H
Long
Lat
H
Long
Lat
DownE
p
EastE
m
NorthE
Vtd
Hd
LatHR
V
td
Longd
HR
V
td
Latd
cos
w
v
u
DownM
EastM
NorthM
BL
BL
BL
BL
BL
BL
BL
BL
BL
V
V
V
CCC
CCC
CCC
w
v
u
_
_
_
3,32,31,3
3,22,21,2
3,12,11,1
DownW
EastW
NorthW
DownE
EastE
NorthE
DownM
EastM
NorthM
V
V
V
V
V
V
V
V
V
_
_
_
_
_
_
_
_
_
DownM
EastM
NorthM
V
V
V
DownE
EastE
NorthE
V
V
V
M
M
Vv
uw
wvuV
/sin
/tan1
1
222
MV
DownE
EastE
NorthE
V
V
V
zBCzBCA
yBCAyBCA
xBCxBCA
TM
TM
TM
,,
,,
,,
DownE
EastE
NorthE
V
V
V
LatHR
V
HR
V
HR
V
EastE
NorthE
EastE
Down
East
North
tan0
0
0
Down
East
North
Down
East
NorthW
L
zW
yW
xW
C
*
*
*
*
*
*
*
zW
yW
xW
*
*
*
1
zW
yW
xW
zW
yW
xW
zW
yW
xW
WLC
Lat
Lat
Down
East
North
sin
0
cos
Down
East
North
Down
East
NorthW
L
zW
yW
xW
C *
*
*
*
*
*
*
zW
yW
xW
*
*
*
1
zW
yW
xW
zW
yW
xW
zW
yW
xW
WLC
Lat
Missile Kinematics Model 1 in Matrix Notation (Spherical Earth)
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
99
SOLO
LBL
BBA
BCG gCT
mF
ma
11
B
BrCrotorB
IBB
BrCrotor
BIBC
BIB
BIBCCTCA
CB
IBII
IIMMI
,,
,,,,1,
BCG
WB
WCG aCa
BIB
BL
LIL
BIBBLBL qqq
2
1
2
1
s
1
CT
CA
M
M
,
,
TBL IqIqC
3434
BIB
BB
BCGa
W
CGa
B
BA
T
F
BLC
BLC W
BC
s
1 BLqBLq
LWa
BLC
IBIWWB zy 11
WM
TWB
BM VCV
L
WB
M
TBL
LE VVCV
s
1
WW
WM
WWWCG
WM
WIW
WM
aVRa
VV
s
1
s
1
WIW
WMV
W
MV
WCGa
23 WBC
WIW B
IB
WMV
B
LCW
BC
WBC
BMV
LEV
LMR
Mee
22M
s
1
s
1
WMV
LWV
WWa
LW
BL
WB
WW aCCa
L
WEL
L
WLW V
td
Vda
2
LWV
L
W
td
Vd
BBr
BBr
,
Missile Kinematics Model 2 in Vector Notation (Spherical Earth)
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
100
SOLO
Missile Kinematics Model 2 in Matrix Notation (Spherical Earth)
rB
rB
r
Crrotor
B
B
B
zzyzxz
yzyyxy
xzxyxx
BB
BB
BB
B
B
B
zzyzxz
yzyyxy
xzxyxx
zBCG
A
yBCG
A
xBCG
A
zzyzxz
yzyyxy
xzxyxx
B
B
B
q
rI
r
q
p
III
III
III
pq
pr
qr
r
q
p
III
III
III
M
M
M
III
III
III
r
q
p
,
1
0
0
0
zBCTzBCA
yBCTyBCA
xBCTxBCA
MM
MM
MM
,,
,,
,,
B
B
B
r
q
p
s
1
B
B
B
r
q
p
B
B
B
r
q
p
4
3
2
1
0
0
0
0
2
1
4
3
2
1
BL
BL
BL
BL
zLzLByLyLBxLxLB
zLzLBxLxLByLyLB
yLyLBxLxLBzLzLB
xLxLByLyLBzLzLB
BL
BL
BL
BL
q
q
q
q
rqp
rpq
qpr
pqr
q
q
q
q
s
1
4
3
2
1
BL
BL
BL
BL
q
q
q
q
4
3
2
1
BL
BL
BL
BL
q
q
q
q
g
C
C
C
T
T
T
mF
F
F
ma
a
a
BL
BL
BL
zB
yB
xB
zBA
yBA
xBA
zB
yB
xB
3,3
3,2
3,111
zBzBA
yByBA
xBxBA
TF
TF
TF
zB
yB
xB
a
a
a
zB
yB
xB
WB
WB
WB
WB
WB
WB
WB
WB
WB
zW
yW
xW
a
a
a
CCC
CCC
CCC
a
a
a
3,32,31,3
3,22,21,2
3,12,11,1
zW
yW
xW
a
a
a
zWM
yWWyWW
yWM
zWWzWW
xWWxWM
V
aar
V
aaq
aaV
zW
yW
xW
a
a
a
W
W
M
r
q
Vs
1
B
WBB
WBB
WBW
BW
BBW
BBW
BW
rCqCpCr
qCrCpCq
3,32,31,3
2,2/3,21,2
s
1
cos0sin
sinsincossincos
cossinsincoscos
23W
BC
WBC
WBC
WBC
WBC
BLC
MV
MV
4
3
2
1
*
1
4
3
2
1
BL
BL
BL
BL
BL
BL
BL
BL
q
q
q
q
q
q
q
q 4
3
2
1
BL
BL
BL
BL
q
q
q
qB
LC
321
412
143
234
3412
2143
1234
BLBLBL
BLBIBL
BLBLBL
BLBLBL
BLBLBLBL
BLBLBLBL
BLBLBLBLB
L
qqq
qqq
qqq
qqq
qqqq
qqqq
qqqq
C
4
3
2
1
BL
BL
BL
BL
q
q
q
q
M
WB
WB
WB
V
C
C
C
w
v
u
3,1
2,1
1,1
w
v
u
s
1
H
Long
Lat
H
Long
Lat
DownW
EastW
NorthW
BL
BL
BL
BL
BL
BL
BL
BL
BL
DownE
EastE
NorthE
V
V
V
w
v
u
CCC
CCC
CCC
V
V
V
_
_
_
_
_
_
3,33,23,1
2,32,22,1
1,31,21,1
DownE
p
EastE
m
NorthE
Vtd
Hd
LatHR
V
td
Longd
HR
V
td
Latd
cos
rr ,
DownE
EastE
NorthE
V
V
V
LatHR
V
HR
V
HR
V
EastE
NorthE
EastE
Down
East
North
tan0
0
0
Down
East
North
Down
East
NorthW
L
zW
yW
xW
C
*
*
*
*
*
*
*
zW
yW
xW
*
*
*
1
zW
yW
xW
zW
yW
xW
zW
yW
xW
WLC
Lat
Lat
Down
East
North
sin
0
cos
Down
East
North
Down
East
NorthW
L
zW
yW
xW
C *
*
*
*
*
*
*
zW
yW
xW
*
*
*
1
zW
yW
xW
zW
yW
xW
zW
yW
xW
WLC
Lat
AIR VEHICLE IN SPHERICAL EARTH ATMOSPHERE
References
SOLO
101
PHAK Chapter 1 - 17http://www.gov/library/manuals/aviation/pilot_handbook/media/
George M. Siouris, “Aerospace Avionics Systems, A Modern Synthesis”, Academic Press, Inc., 1993
R.P.G. Collinson, “Introduction to Avionics”, Chapman & Hall, Inc., 1996, 1997, 1998
Ian Moir, Allan Seabridge, “Aircraft Systems, Mechanical, Electrical and AvionicsSubsystem Integration”, John Wiley & Sons, Ltd., 3th Ed., 2008
Fighter Aircraft Avionics
Ian Moir, Allan Seabridge, “Military Avionics Systems”, John Wiley & Sons, LTD., 2006
References (continue – 1)
SOLO
102
Fighter Aircraft Avionics
S. Hermelin, “Air Vehicle in Spherical Earth Atmosphere”
S. Hermelin, “Airborne Radar”, Part1, Part2, Example1, Example2
S. Hermelin, “Tracking Systems”
S. Hermelin, “Navigation Systems”
S. Hermelin, “Earth Atmosphere”
S. Hermelin, “Earth Gravitation”
S. Hermelin, “Aircraft Flight Instruments”
S. Hermelin, “Computing Gunsight, HUD and HMS”
S. Hermelin, “Aircraft Flight Performance”
S. Hermelin, “Sensors Systems: Surveillance, Ground Mapping, Target Tracking”
S. Hermelin, “Air-to-Air Combat”
References (continue – 2)
SOLO
103
Fighter Aircraft Avionics
S. Hermelin, “Spherical Trigonometry”
S. Hermelin, “Modern Aircraft Cutaway”
104
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 –
Stanford University1983 – 1986 PhD AA
105
SOUND WAVESSOLO
SupersonicV > a
SubsonicV < a
a t a t
V tV t
M
1sin 1
Soundwaves
Machwaves
Disturbances propagate by molecular collision, at the sped of sound a,along a spherical surface centered at the disturbances source position.
The source of disturbances moves with the velocity V.
- when the source moves at subsonic velocity V < a, it will stay inside the family of spherical sound waves.
-when the source moves at supersonic velocity V > a, it will stay outside the family of spherical sound waves. These wave fronts form a disturbance
envelope given by two lines tangent to the family of spherical sound waves. Those lines are called Mach waves, and form an angle μ with the disturbance
source velocity:a
VM
M
&
1sin 1
106
SOUND WAVESSOLO
Sound Wave Definition: p
p
p p
p1
2 1
1
1
2 1
2 1
2 1
p p p
h h h
For weak shocks
up
1
2
11
11
1
11
11
2
12
1
1uuuuuu
)C.M.(
ppuuupuupu
11
111122111
211
)C.L.M.(
21
au 1
1p
1
1T
1e
112 uuu
112 ppp
112
112 TTT
112 eee
SOUND
WAVE
Since the changes within the sound wave are small, the flow gradients are small.Therefore the dissipative effects of friction and thermal conduction are negligibleand since no heat is added the sound wave is isotropic. Since
au 1
s
pa
2valid for all gases
107
SPEED OF SOUND AND MACH NUMBERSOLO
21
au 1
1p
1
1T
1e
112 uuu
112 ppp
112
112 TTT
112 eee
SOUNDWAVE
Speed of Sound is given by
0
ds
pa
RTp
C
C
T
dT
R
C
pT
dT
R
C
d
dp
dR
T
dTCds
p
dpR
T
dTCds
v
p
v
p
dsv
p
00
0
but for an ideal, calorically perfect gas
pRTa
TChPerfectyCaloricall
RTpIdeal
p
The Mach Number is defined asRT
u
a
uM
1
2
1
1
111
a
a
T
T
p
pThe Isentropic Chain:
a
ad
T
Tdd
p
pdsd
1
2
10
108
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q 0 0,
Mach Number Relations (1)
122
22
1
21
22
222
21
221
22
2
222
1
1
21
1222
2
11
1
22221
211
2211
2
1
2
12
1
2
1
*12
1
2
1
12
1
14..
...
..
uuu
a
u
a
uaa
uaaau
h
au
h
aEC
uuu
p
u
p
pupuMLC
uuMCp
a
Field Equations:
1222
2
11
2
2
1
2
1
2
1
2
1uuu
u
au
u
a
u u a1 22
u
a
u
aM M1 2
1 21 1
Prandtl’s Relation
u
p
T
e
u
p
T
e
11
q
1
1
1
1
1
2
2
2
2
2
1 2
2
1
2
11
2
1
2
1
2
1
21
2
12122
21
12
uu
auuuua
uu
uu
Ludwig Prandtl(1875-1953)
109
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q 0 0,
Mach Number Relations (2)
M
MM
M
M
M
M
MM
22
22
1
12
12
12
12
12
21
1
2
1 1
2
11
1 21
2 1 2
1 1 1 1 1
12
or
M
M
M
M
MH H
A A
2
12
12
12
121 2
1 21
1
21
2
2
1
11
2
12
11
2
1
1
2
12
1 2
12
2 12 1
2
12
1 2 1
1 2
A A u
u
u
u u
u
aM
M
M
u
p
T
e
u
p
T
e
11
q
1
1
1
1
1
2
2
2
2
2
1 2
110
NORMAL SHOCK WAVESSOLO
Normal Shock Wave ( Adiabatic), Perfect Gas
G Q 0 0,
Mach Number Relations (3)
p
p
up
u
u
u
a
MM
MM
M M
M
2
1
12
1
1
2
1
12
12
1
2
12 1
2
12 1
2 12
12
12
1 1 1 1
1 11 2
11
1 1 2
1
or
(C.L.M.)
p
pM2
1121
2
11
h
h
T
T
p
pM
M
M
a
a
h C T p R Tp2
1
2
1
2
1
1
212 1
2
12
2
1
12
11
1 2
1
s s
R
T
T
p
pM
M
M2 1 2
1
12
1
1
12
1
112
12
1
12
11
1 2
1
ln ln
s s
RM M
M2 1
1 1
2 12 3
2
2 12 41
2 2
3 11
2
11
Shapiro p.125
u
p
T
e
u
p
T
e
11
q
1
1
1
1
1
2
2
2
2
2
1 2
111
STEADY QUASI ONE-DIMENSIONAL FLOWSOLO
STAGNATION CONDITIONS
)C.E.( constuhuh 222
211 2
1
2
1
The stagnation condition 0 is attained by reaching u = 0
2
/
21202020
2
11
12
12
122
12
MTR
u
Tc
u
T
T
c
uTTuhh
TRa
auM
Rc
pp
Tch pp
Using the Isentropic Chain relation, we obtain:
2
10102000
2
11 M
p
p
a
a
h
h
T
T
Steady , Adiabatic + Inviscid = Reversible, , q Q 0 0, ~ ~ 0
G 0 t
0
SOLO
112
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SOLO
113
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SOLO
114
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SOLO
115
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SOLO
116
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SOLO
117
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SOLO
118
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119
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SOLO
120
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121